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2 173 ABSTRACT The evolution of increasing turbine inlet temperature has led to the necessity of full-coverage film cooling for the first turbine vane and blade. A new approach using high speed wall jets for blade cooling has been proposed by the authors. In this paper a 2-D upwind-biased Navier-Stokes code is used to calculate the aerodynamic behaviour of these surface coolant jets in a linear cascade. Special emphasis is put on the investigation of the coolant flow around the leading edge. Various numerical results concerning the leading edge flow are presented in detail, showing a strong tendency of these jets to bend towards convex surfaces. Finally, Schlieren pictures taken at the Institute s transonic cascade on a 140mm chord length blade are presented for comparison. NOMENCLATURE Symbols: A + =26 l L M P P * T T 0 u,v,w x y y + δ κ constant for the turbulence model mixing length chord length Mach number total pressure coolant/total pressure main flow critical pressure coolant/total pressure main flow static temperature total temperature cartesian velocity components chordwise coordinate minimum dis tance to any wall nondimensiomal distance from the wall boundary-layer thickness ratio of specific heats ω vorticity µ t eddy viscosity µ 0 laminar viscosity at stagnation Indexes: c coolant m main flow 1 at inflow boundary INTRODUCTION The objective of this work is to investigate underexpanded jets for a novel turbine blade film cooling system in a linear cascade, especially their aerodynamic behaviour around the leading edge. Since these underexpanded jets have a strong tendency to bend towards the surface a higher cooling efficiency is expected due to the improved cooling film attachment. The effect of air flows bending around curved surfaces is generally connected with the name "Coanda-effect" and originally describes subsonic flows attaching to the surface by their turbulent structure (e.g. Fernholz and Wille, 1971; Ameria and Dybbs, 1993). Several experiments with underexpanded jets bending towards a convex surface have been performed and published (Gregory-Smith and Gilchrist, 1987; Gilchrist and Gregory-Smith, 1988; Gregory-Smith and Hawkins, 1991; Gregory-Smith and Senoir, 1994). These curved supersonic jets have a complex structure with compression and expansion waves in the underexpanded core of the jet, with an outer free shear layer of the jet and a boundary layer towards the surface. Since these underexpanded jets bend towards convex surfaces it has been suggested (Jericha et al., 1995) that its use might be considered for high temperature turbine blade film cooling. Cooling slits are suggested instead of holes in order to preserve this major property and also to cover a large area of the blade.

3 174 A preliminary investigation was performed with an acrylic glass model of a cooling slit (Woisetschläger et. al. 1995) which then provided the basis for a cooling slit design inside the linear blade cascade at the Institute of Thermal Turbomachinery and Machine Dynamics, TU Graz (s. fig. 1). CO 2 window section cooler compressor station Fig. 1: Linear cascade, test section The test section was designed to encompass a combined cycle proposed by Jericha et al. (1995) using high pressure steam as cooling medium for the high temperature gas turbine. In this case there would be no need of additional blowing energy. Since it is not possible to run the linear cascade test rig with the same flow temperatures as a real turbine an aerodynamically similar test section was designed using CO 2 as cooling flow medium. A detailed description of the measurement techniques has been given by Woisetschläger et al. (1996). In the work presented here a 2-D Navier-Stokes code, based on the unsteady Euler code of Sanz et al. (1995), is used to demonstrate the effects described above and to investigate the influence of the supersonic cooling flow on the main flow field. For comparison with experimental data the blade profile and the slit geometry of the test section and the respective pressure ratios and temperatures have been investigated. Since this Navier-Stokes code is limited to single-phase flow problems, the CO 2-flow in the cooling slit is calculated separately from the main flow. The respective results for velocity, pressure and density in the throat of the slit, where the average Mach number becomes unity, are then used as inflow boundary conditions for the main flow. NUMERICAL METHOD The unsteady, Reynolds averaged Navier Stokes equations are treated in conservative form and discretized in space using a cell-centred finite volume formulation with a quadrilateral structured cell system and in time using the Euler implicit method. air Inviscid Fluxes On each cell boundary the inviscid fluxes are evaluated by an upwind scheme based on the approximate Riemann solver of Roe (1981). To raise the order of spatial accuracy the inviscid fluxes are evaluated using a 3rd order accurate TVD scheme. The TVD scheme is based on the MUSCL type of an upwind scheme which consists of a projection stage and an evolution stage. In the projection stage, left and right states at each cell interface are determined by extrapolating the cell-centred values of the conservative variables toward the cell interface according to the third-order upwind-biased scheme with a differentiable limiter (Anderson et al., 1985). In the evolution stage the inviscid flux is evaluated by solving the Riemann problem between left and right states using Roe's approximate Riemann solver. The implicit portions of the inviscid flux vectors are simplified by defining an approximation to the true linearization. Therefore, only a first order accurate scheme is considered for the implicit side whereas the full high-accuracy scheme on the explicit side is used (Sanz et al. 1995). Viscous Fluxes In order to construct the numerical viscous flux vector at the cell interfaces it is necessary to evaluate first-order derivatives of the velocity components and the sound speed, which is done in a centraldifferencing manner, using Green's theorem (e.g. Furukawa et al. 1991). The time linearization of the viscous flux vector is performed by applying the thin-layer approximation for the implicit side of the equations. Turbulence Model A two-layer algebraic model based on the mixing length concept is used for turbulence closure. In the near wall region the mixing length is computed using the Prandtl-Van Driest formula while in the outer region and in the wake it is kept constant to a fixed fraction of the shear layer thickness δ, according to the standard relation (Kwon et al, 1988). l outer = δ In the present work an algebraic criterion, which resembles the features of the Baldwin Lomax (1978) model, but which implicitly introduces a cut-off criterion for the vorticity field based on the distance from the wall, is used to estimate the boundary layer thickness (Arnone et al. 1996). If y denotes the distance normal to the wall, the value y max at which the function y + y 1 + G( y ) = y e A dy y ω 1 0 reaches its maximum is assumed as turbulent length scale. The boundary layer thickness is then obtained from the relationship: δ= y max

4 175 Boundary Conditions In cascade-like configurations, there are four different types of boundaries: inlet, outlet, solid wall and periodicity. In the present cell-centred scheme, phantom cells are used to handle all boundaries. According to the theory of characteristics, flow angle, total pressure, total temperature and isentropic relations are used at the subsonic axial inlet, whereas all variables are prescribed at the supersonic inlet. At the subsonic axial outlet the average value of the static pressure is prescribed, density and velocity components are extrapolated, whereas all variables are extrapolated at supersonic axial outlet. On solid walls, the pressure is extrapolated from the interior points and the no-slip adiabatic condition is used to compute density and total energy. The periodicity is imposed by setting periodic phantom cell values. GEOMETRY, GRID SYSTEM The blade profile with its cooling slits as it is built in the test section and the main design parameters are given in fig. 2. All calculations presented in this paper concentrate on the rather difficult situation near cooling slit #1, slits #2 and #3 are not subject of these investigations. Fig. 3.a: Periodic C-type grid for the main flow without cooling jets (251 x 30 nodes) blade chord length 143 mm blade pitch 98.6 mm cooling slit throat 0.2 mm design inlet angle * 55 design outlet angle * 28 * measured from circumferential direction cooling slit # cooling slit # Fig. 3.b: Periodic C-type grid for the main flow with cooling jets (274 x 40 nodes) cooling slit #1 x chord length L = 142mm Fig. 2: Geometry of the blade and cooling slits An algebraic mesh generator, developed by Gehrer et al. (1996), based on Bézier curves and Bézier surfaces was applied to generate a periodic C-type grid for the main flow field, which is shown in fig. 3. Two different grids were used to calculate the linear cascade flow. Fig. 3.a shows the grid for the calculations without coolant which consists of 251 x 30 nodes. A refined version of this grid (s. figs. 3.b, 3.c), consisting of 274 x 40 nodes and including the throat of the cooling slit, is used to calculate the main flow with coolant. Fig. 3.c: Area around cooling slit Fig. 3.d: Mesh for the flow inside the cooling slit (30x27 nodes)

6 177 even in the region of the highly curved leading edge radius and even in a direction opposite to the incoming flow. The numerical results given in table 2 show a slight reduction of both inlet mass flow and exit flow angle, which indicates that the high speed cooling jet enhances the overall blade vortex. Fig. 4.d: Pressure contours for case 1 Results for the flow inside the cooling slit (e.g. case2) In fig. 5 computational results for the CO 2 - flow inside the cooling slit (e.g. case 2) are presented. Fig. 5.a: Cooling slit: Mach number contours Fig. 6.a: Density contours for case 2 u[m/s] u (NS-calculation) M= y[m] Fig. 5.b: Cooling slit: Velocity distribution across throat In fig. 5.a the Mach number contours illustrate the growth of the boundary layer inside the slit, which is, due to the relatively low Reynolds number (s. table 2) of about 10 4, rather pronounced. In fig. 5.b, the velocity distribution across the throat of the slit is displayed. As mentioned above, the calculated velocity, density and pressure distributions across the throat (see also fig. 3.c) are prescribed as supersonic inflow boundary for the main flow for cases 2 and 3. However, the result for the supersonic part (throat to outlet in fig. 3.d) is not used for the main flow calculation. It should be noted that also the subsonic boundary layer region in the throat was treated as supersonic inflow boundary for the main flow. Results for the main flow with cooling jet (cases 2 and 3) Regarding the flow with coolant, the respective density contours in figs. 6.a and 6.b clearly demonstrate that it appears possible to keep the cooling layer close to the surface of the blade, Fig. 6.b: Density contours for case 3 A more detailed view of the domain of interest in fig. 8 illustrates that the outflow of supersonic cooling medium from the slit forms a cooling layer around the leading edge of the blade. Especially the Mach number distributions in figs. 8.e and 8.f show that the jet expands to transonic velocities and, at the same time, due to the effect of continuous expansion, the film adheres very closely to the surface. This expansion can also be noticed as a pressure drop, which is clearly visible in the pressure contours.(s. figs. 8.a and 8.b). For purpose of comparison, the Schlieren visualisation of the leading edge flow is presented in fig.7 which can be regarded as a qualitative verification of this effect.

8 179 Mis no filmcooling P=2 P= X/L 2 Suction Side 1 Pressure Side Fig. 9.a: Isentropic Mach number distribution in the area near the leading edge The isentropic Mach number distribution in the area around the leading edge (s. fig. 9.a) shows a significant peak shortly after the coolant is injected at the pressure side of the blade. This means that the jets definitely accelerates the main flow (in flow direction of the jet) around the leading edge, which is also clearly visible in fig. 9.b, where the respective velocity profiles are given. The locations of gridlines used to evaluate these profiles are displayed in figs. 8.e and 9.a. T/T no filmcooling P=2 P=3 Pressure Side Suction Side tangential velocity [m/s], scale: 1 increment = 50 m/s y/l (normal to the wall) P = 3 no cooling P = 2 Fig. 9.b: Velocity profiles in the area of the leading edge. The respective locations are denoted by the numbers These positions are also shown in figs. 8.e and 9.a Fig. 10: Predicted temperature at the wall, normalised with the main flow total inlet temperature Finally, the predicted temperature distribution at the (adiabatic) wall, normalised by the main flow total inlet temperature is presented (s. fig. 10). It should be kept in mind, that the results from the CO 2 flow calculations (total temperature, CO 2 : K) inside the cooling slit were transferred to the main flow field (air), keeping the same pressure, velocity and density, which results in a total temperature of about 193 K. Under these conditions, the calculations predict, that the cold cooling jet fluid remains along the whole suction side, its temperature rises due to friction, heat transfer and mixing with the main flow. Finally, it can be noticed, t hat all numerical results for case 3 are qualitatively similar to the results for case 2, which means that this range of pressure ratio P seems possible for a strong adherence between this type of cooling film and the blade surface. Investigations, presented by Gilchrist and Gregory-Smith (1988) and Woisetschläger et al. (1995), show that further increase of pressure ratio P leads to a low speed recirculation zone between jet and surface in the boundary layer which then causes a jet "breakaway" from the surface. CONCLUSION Underexpanded cooling films have a strong tendency to bend towards the surface which makes them interesting especially for cooling of the leading edge area. In this work, 2D Navier Stokes calculations were used to demonstrate, that transonic pressure ratios X/L

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